About: Mercer's theorem

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In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel.

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rdfs:label	Satz von Mercer (de) Théorème de Mercer (fr) Mercer's theorem (en)
rdfs:comment	Der Satz von Mercer ist eine mathematische Aussage aus dem Teilgebiet der Funktionalanalysis. Er ist benannt nach dem Mathematiker James Mercer und besagt, dass der Integralkern eines positiven, selbstadjungierten Integraloperators als konvergente Reihe über seine Eigenwerte und Eigenvektoren dargestellt werden kann. (de) En mathématiques et plus précisément en analyse fonctionnelle, le théorème de Mercer est une représentation d'une fonction symétrique de type positif par le carré d'une série convergente de produits de fonctions. Ce théorème est l'un des résultats phares de James Mercer. C'est un outil théorique important dans la théorie des équations intégrales. Il est aussi utilisé dans la théorie hilbertienne des processus stochastiques (voir (en) et Transformée de Karhunen-Loève). (fr) In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel. (en)
dcterms:subject	Theorems in functional analysis
Wikipage page ID	303990 (xsd:integer)
Wikipage revision ID	1116343803 (xsd:integer)
Link from a Wikipage to another Wikipage	Borel algebra David Hilbert Definite bilinear form Richard Courant Uniform norm Integral equation Theorems in functional analysis Mathematics Spectral theory of compact operators Symmetric matrix Eigenfunction Equicontinuous Function (mathematics) Gramian matrix Theorem Methoden der mathematischen Physik Representer theorem Linear operator Lp space Compact operator on Hilbert space Fubini's theorem Functional analysis Kernel method Orthonormal basis Spectral theory James Mercer (mathematician) Non-negative Hilbert–Schmidt integral operator Hilbert–Schmidt operator Konrad Jörgens Hilbert space First-countable Karhunen–Loève theorem Trace class Positive-definite kernel Positive-semidefinite matrix Spectral theorem Injective function Real number Kernel trick Square-integrable function Unit ball Stochastic process Orthonormal set Ascoli's theorem Eigenvalues Compact Hausdorff space
sameAs	Mercer's theorem Mercer's theorem Mercer's theorem Mercer's theorem Mercer's theorem Mercer's theorem
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Citation_needed dbt:Harv dbt:Reflist dbt:Functional_analysis
id	p/m063440 (en)
title	Mercer theorem (en)
has abstract	Der Satz von Mercer ist eine mathematische Aussage aus dem Teilgebiet der Funktionalanalysis. Er ist benannt nach dem Mathematiker James Mercer und besagt, dass der Integralkern eines positiven, selbstadjungierten Integraloperators als konvergente Reihe über seine Eigenwerte und Eigenvektoren dargestellt werden kann. (de) En mathématiques et plus précisément en analyse fonctionnelle, le théorème de Mercer est une représentation d'une fonction symétrique de type positif par le carré d'une série convergente de produits de fonctions. Ce théorème est l'un des résultats phares de James Mercer. C'est un outil théorique important dans la théorie des équations intégrales. Il est aussi utilisé dans la théorie hilbertienne des processus stochastiques (voir (en) et Transformée de Karhunen-Loève). (fr) In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. This theorem, presented in, is one of the most notable results of the work of James Mercer (1883–1932). It is an important theoretical tool in the theory of integral equations; it is used in the Hilbert space theory of stochastic processes, for example the Karhunen–Loève theorem; and it is also used to characterize a symmetric positive semi-definite kernel. (en)
gold:hypernym	Representation
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foaf:isPrimaryTopicOf	wikipedia-en:Mercer's_theorem
is Link from a Wikipage to another Wikipage of	List of functional analysis topics Jovan Karamata List of University of Manchester people Regularized least squares Bergman kernel String kernel Empirical orthogonal functions Functional data analysis Representer theorem Compact operator on Hilbert space Kernel method Axis–angle representation James Mercer (mathematician) Gram matrix Isomap

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